If $V$ is a Hermitian matrix ($\bar{V}= V^T$), $V$ is of size $2n \times2n$ and
$
V = \begin{pmatrix}
0 & A \\
\bar{A}^T & B
\end{pmatrix}
$ and $\det V \neq 0$ then $\sigma(V) = 0$.
\end{lemma}
\begin{definition}
A Hermitian form $V$ is metabolic if $V$ has structure
$\begin{pmatrix}
0 & A\\
\bar{A}^T & B
\end{pmatrix}$ with half-dimensional null-space.
\end{definition}
\noindent
In other words: non-degenerate metabolic hermitian form has vanishing signature.\\
We note that $\det(S + S^T)\neq0$. Hence $\det((1- t) S +(1-\bar{t})S^T)$ is not identically zero on $S^1$, so it is non-zero except possibly at finitely many points. We apply the Lemma \ref{lem:metabolic}. \\
Let $t \in S^1\setminus\{1\}$. Then:
\begin{align*}
&\det((1 - t) S + (1 - \bar{t}) S^T) =
\det((1 - t) S + (t\bar{t} - \bar{t}) S^T) =\\
&\det((1 - t) (S - \bar{t} - S^T)) =
\det((1 -t)(S - \bar{t} S^T)).
\end{align*}
As $\det(S + S^T)\neq0$, so $S -\bar{t}S^T \neq0$.
\end{proof}
\begin{corollary}
If $K \sim K^\prime$ then for all but finitely many $t \in S^1\setminus\{1\}: \sigma_K(t)=\sigma_{K^\prime}(t)$.
\end{corollary}
\begin{proof}
If $ K \sim K^\prime$ then $K \# K^\prime$ is slice.
\[
\sigma_{-K^\prime}(t) = -\sigma_{K^\prime}(t)
\]
\\??????????????\\
The signature give a homomorphism from the concordance group to $\mathbb{Z}$.\\
??????????????????\\
Remark: if $t \in S^1$ is not algebraic over $\mathbb{Z}$, then $\sigma_K(t)\neq0$
(we can is the argument that $\mathscr{C}\longrightarrow\mathbb{Z}$ as well).
\dim (\ker (H_1, Y) \longrightarrow H_1(\Omega)) = n + g\\
Y = X \sum\Sigma
\end{align*}
\noindent
If $\alpha, \beta\in\ker(H_1(\Sigma\longrightarrow H_1(\Omega))$, then ${\Lk(\alpha, \beta^+)=0}$.
\begin{corollary}
If $t$ is nota ???? of $\det$ ????
then $\vert\sigma_K(t)\vert\leq2g$.\\
\end{corollary}
\noindent
If there exists cobordism of genus $g$ between $K$ and $K^\prime$ like shown in Figure \ref{fig:genus_2_bordism}, then $K \#-K^\prime$ bounds a surface of genus $g$ in $B^4$.
The (smooth) four genus $g_4(K)$ is the minimal genus of the surface $\Sigma\in B^4$ such that $\Sigma$ is compact, orientable and $\partial\Sigma= K$.
\end{definition}
\noindent
Remark: $3$ - genus is additive under taking connected sum, but $4$ - genus is not.